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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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c^a + a = 2^b
Havu   17
N 2 minutes ago by Royal_mhyasd
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
17 replies
Havu
May 10, 2025
Royal_mhyasd
2 minutes ago
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How many friends can sit in that circle at most?
Arytva   3
N 17 minutes ago by imagien_bad

A group of friends sits in a ring. Each friend picks a different whole number and holds a stone marked with it. Then they pass their stone one seat to the right so everyone ends up with two stones: one they made and one they received. Now they notice something odd: if your original number is $x$, your right-neighbor’s is $y$, and the next person over is $z$, then for every trio in the circle they see

$$ x + z = (2 - x)\,y. $$
They want as many friends as possible before this breaks (since all stones must stay distinct).

How many friends can sit in that circle at most?
3 replies
Arytva
Today at 10:00 AM
imagien_bad
17 minutes ago
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A circle tangent to the circumcircle
kosmonauten3114   0
21 minutes ago
Source: My own (well-known?)
Let $\triangle{ABC}$ be a scalene triangle with incircle $\odot(I)$.
Let $\odot(O_A)$ be the circle tangent to $\odot(I)$ and passing through $B$ and $C$, and denote by $A_B$, $A_C$ the second intersection points of $\odot(O_A)$ and $AB$, $AC$, resp. Define $B_C$, $B_A$, $C_A$, $C_B$ cyclically.
Let $\odot(O')$ be the circle internally tangent to $\odot(AA_BA_C)$, $\odot(BB_CB_A)$, $\odot(CC_AC_B)$.

Prove that $\odot(O')$ is tangent to $\odot(ABC)$.
0 replies
kosmonauten3114
21 minutes ago
0 replies
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Subset with lower bound on sum
mkultra42   1
N 27 minutes ago by Gliese
Given $2n$ non-negative integers whose sum is $3n$. Prove that there exists $n$ of them whose sum is at least $2n$.
1 reply
mkultra42
an hour ago
Gliese
27 minutes ago
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inequality in 2021 China Second Round A2
EthanWYX2009   2
N 44 minutes ago by phi22_7
Source: 2021 China Second Round A2
Given $n\geq 2$, $a_1$, $a_2$, $\cdots$, $a_n\in\mathbb {R}$ satisfy
$$a_1\geqslant a_2\geqslant \cdots \geqslant a_n\geqslant 0,a_1+a_2+\cdots +a_n=n.$$Find the minimum value of $a_1+a_1a_2+\cdots +a_1a_2\cdots a_n$.
2 replies
EthanWYX2009
Feb 20, 2023
phi22_7
44 minutes ago
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Beautiful geo but i cant solve this
phonghatemath   0
an hour ago
Source: homework
Given triangle $ABC$ inscribed in $(O)$. Two points $D, E$ lie on $BC$ such that $AD, AE$ are isogonal in $\widehat{BAC}$. $M$ is the midpoint of $AE$. $K$ lies on $DM$ such that $OK \bot AE$. $AD$ intersects $(O)$ at $P$. Prove that the line through $K$ parallel to $OP$ passes through the Euler center of triangle $ABC$.

Sorry for my English!
0 replies
phonghatemath
an hour ago
0 replies
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Points on a lattice path lies on a line
navi_09220114   4
N an hour ago by atdaotlohbh
Source: TASIMO 2025 Day 1 Problem 3
Let $S$ be a nonempty subset of the points in the Cartesian plane such that for each $x\in S$ exactly one of $x+(0,1)$ or $x+(1,0)$ also belongs to $S$. Prove that for each positive integer $k$ there is a line in the plane (possibly different lines for different $k$) which contains at least $k$ points of $S$.
4 replies
navi_09220114
May 19, 2025
atdaotlohbh
an hour ago
J
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Functional equation
MuradSafarli   1
N an hour ago by MuradSafarli
Source: Germany 2007
Let \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) be a function such that for all positive rational numbers \( x, y \), the following equation holds:
\[ f\left(\frac{f(x)}{y f(x) + 1}\right) = \frac{x}{x f(y) + 1}. \]
1 reply
MuradSafarli
an hour ago
MuradSafarli
an hour ago
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Directed edge chromatic numbers over a tournament
v_Enhance   33
N an hour ago by yayyayyay
Source: USA January TST for 56th IMO, Problem 2
A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors. Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices.

Proposed by Po-Shen Loh
33 replies
v_Enhance
Mar 22, 2015
yayyayyay
an hour ago
J
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2-var inequality
sqing   11
N an hour ago by ytChen
Source: Own
Let $ a,b> 0 ,a^3+ab+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq 8$$$$ (a^2+b^2)(a+1)(b+1) \leq 8$$Let $ a,b> 0 ,a^3+ab(a+b)+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$
11 replies
sqing
Yesterday at 1:35 PM
ytChen
an hour ago
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Prove angles are equal
BigSams   52
N an hour ago by zuat.e
Source: Canadian Mathematical Olympiad - 1994 - Problem 5.
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
52 replies
1 viewing
BigSams
May 13, 2011
zuat.e
an hour ago
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Angles in a triangle with integer cotangents
Stear14   2
N 2 hours ago by Stear14
In a triangle $ABC$, the point $M$ is the midpoint of $BC$ and $N$ is a point on the side $BC$ such that $BN:NC=2:1$. The cotangents of the angles $\angle BAM$, $\angle MAN$, and $\angle NAC$ are positive integers $k,m,n$.
(a) Show that the cotangent of the angle $\angle BAC$ is also an integer and equals $m-k-n$.
(b) Show that there are infinitely many possible triples $(k,m,n)$, some of which consisting of Fibonacci numbers.
2 replies
Stear14
May 21, 2025
Stear14
2 hours ago
J
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Parallelograms and concyclicity
Lukaluce   33
N 2 hours ago by HamstPan38825
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
33 replies
Lukaluce
Apr 14, 2025
HamstPan38825
2 hours ago
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IMO Shortlist 2013, Number Theory #4
lyukhson   30
N 2 hours ago by Martin2001
Source: IMO Shortlist 2013, Number Theory #4
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
30 replies
lyukhson
Jul 10, 2014
Martin2001
2 hours ago
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the four altitudes are concurrent
Fermat -Euler   6
N Sep 19, 2022 by levifb
Source: brazil 2001
An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
6 replies
Fermat -Euler
Sep 5, 2005
levifb
Sep 19, 2022
J
the four altitudes are concurrent
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Reveal topic tags
geometry proposed
geometry
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Source: brazil 2001
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Fermat -Euler
444 posts
Fermat -Euler
#1 h  Sep 5, 2005, 9:09 AM • 2 Y
Y by Adventure10, Mango247
An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
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mecrazywong
606 posts
mecrazywong
#2 h  Sep 5, 2005, 2:06 PM • 1 Y
Y by Adventure10
This is well known. The lines concur at the anticenter of quadrilateral $ABCD$.
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Arne
3660 posts
Arne
#3 h  Sep 5, 2005, 2:11 PM • 2 Y
Y by vsathiam, Adventure10
mecrazywong wrote:
The lines concur at the anticenter of quadrilateral $ABCD$.

More information here:

http://www.cut-the-knot.org/Curriculum/Geometry/Brahmagupta2.shtml
http://www.pballew.net/cycquad.html

etcetera
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vsathiam
201 posts
vsathiam
#4 h  Aug 19, 2017, 9:12 PM • 2 Y
Y by Adventure10, Mango247
Or you can complex bash this :)

Once you've proved the cyclic part, you can use that to disprove noncyclic quads.
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jayme
9801 posts
jayme
#5 h  Aug 30, 2017, 1:19 PM • 4 Y
Y by vsathiam, levifb, Adventure10, Mango247
Dear Mathlinkers,
we can speak instead altitude, maltitude...
see

http://jl.ayme.pagesperso-orange.fr/Docs/A%20propos%20de%20l'anticentre%20d'un%20quadrilatere%20cyclique.pdf

Sincerely
Jean-Louis
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jestrada
520 posts
jestrada
#6 h  Aug 30, 2017, 2:09 PM • 1 Y
Y by Adventure10
The midpoints of a quadrilateral form a parallelogram (even better known). Let $X$ be the center of the parallelogram. Rotate $ABCD$ 180 degrees through $X$.
If $ABCD$ is cyclic, let $O$ be the circumcenter. The lines from $O$ to the midpoints are the perpendicular bisectors of the respective sides; the rotation takes the midpoint of each side to the midpoint of the opposite side, and the images of the lines from $O$ to the midpoints are precisely the altitudes of $ABCD$, which meet in the image of $O$. Likewise if the altitudes are concurrent, the rotation takes the altitudes to the perpendicular bisectors of the respective sides, and if these are concurrent then $ABCD$ is cyclic.
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levifb
61 posts
levifb
#7 h  Sep 19, 2022, 5:29 PM
Y by
Just consider the homothety with center in the center of the parallelogram formed by the midpoints of the sides, and radius $-1$. Notice that the altitudes are taken in the perpendicular bisectors of the sides of the quadrilateral. Thus, they concur if and only if this lines meet at an unic point, i.e, ABCD is cyclic.
This post has been edited 1 time. Last edited by levifb, Sep 23, 2022, 10:49 PM
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